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<p class="purpose">Mathematics  Mathematical functions.</p>

<ul class="toc"><li><a href="mthmt.html#SP1">&#167;1. Unsigned integer comparison</a></li><li><a href="mthmt.html#SP2">&#167;2. Fully random word</a></li><li><a href="mthmt.html#SP3">&#167;3. Integer roots</a></li><li><a href="mthmt.html#SP4">&#167;4. Real number support</a></li></ul><hr class="tocbar">

<p class="commentary firstcommentary"><a id="SP1" class="paragraph-anchor"></a><b>&#167;1. Unsigned integer comparison.</b>Comparison of integers is normally signed, that is, treating the word as
a twos-complement signed number, so that <span class="extract"><span class="extract-syntax">$FFFF</span></span> is less than <span class="extract"><span class="extract-syntax">0</span></span>, for
instance. If we want to construe words as being unsigned integers, or as
addresses, we need to compare them with the following routine, which returns
1 if \(x&gt;y\), 0 if \(x=y\) and \(-1\) if \(x&lt;y\).
</p>

<pre class="displayed-code all-displayed-code code-font">
<span class="plain-syntax">[ </span><span class="identifier-syntax">UnsignedCompare</span><span class="plain-syntax"> </span><span class="identifier-syntax">x</span><span class="plain-syntax"> </span><span class="identifier-syntax">y</span><span class="plain-syntax"> </span><span class="identifier-syntax">u</span><span class="plain-syntax"> </span><span class="identifier-syntax">v</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">x</span><span class="plain-syntax"> == </span><span class="identifier-syntax">y</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">0</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">x</span><span class="plain-syntax"> &lt; </span><span class="constant-syntax">0</span><span class="plain-syntax"> &amp;&amp; </span><span class="identifier-syntax">y</span><span class="plain-syntax"> &gt;= </span><span class="constant-syntax">0</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">1</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">x</span><span class="plain-syntax"> &gt;= </span><span class="constant-syntax">0</span><span class="plain-syntax"> &amp;&amp; </span><span class="identifier-syntax">y</span><span class="plain-syntax"> &lt; </span><span class="constant-syntax">0</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> -1;</span>
<span class="plain-syntax">    </span><span class="identifier-syntax">u</span><span class="plain-syntax"> = </span><span class="identifier-syntax">x</span><span class="plain-syntax">&amp;~</span><span class="identifier-syntax">WORD_HIGHBIT</span><span class="plain-syntax">; </span><span class="identifier-syntax">v</span><span class="plain-syntax">= </span><span class="identifier-syntax">y</span><span class="plain-syntax">&amp;~</span><span class="identifier-syntax">WORD_HIGHBIT</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">u</span><span class="plain-syntax"> &gt; </span><span class="identifier-syntax">v</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">1</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">return</span><span class="plain-syntax"> -1;</span>
<span class="plain-syntax">];</span>
</pre>
<p class="commentary firstcommentary"><a id="SP2" class="paragraph-anchor"></a><b>&#167;2. Fully random word.</b>This should be our best try at a single word consisting of 16 uniformly
random bits. <span class="extract"><span class="extract-syntax">random($100)-1</span></span> is a fully random byte.
</p>

<pre class="displayed-code all-displayed-code code-font">
<span class="plain-syntax">[ </span><span class="identifier-syntax">FullyRandomWord</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">return</span><span class="plain-syntax"> (</span><span class="reserved-syntax">random</span><span class="plain-syntax">(</span><span class="constant-syntax">$100</span><span class="plain-syntax">)-1)*</span><span class="constant-syntax">$100</span><span class="plain-syntax"> + (</span><span class="reserved-syntax">random</span><span class="plain-syntax">(</span><span class="constant-syntax">$100</span><span class="plain-syntax">)-1);</span>
<span class="plain-syntax">];</span>
</pre>
<p class="commentary firstcommentary"><a id="SP3" class="paragraph-anchor"></a><b>&#167;3. Integer roots.</b>We are unable to provide VM support for taking square or cube roots rapidly:
</p>

<pre class="displayed-code all-displayed-code code-font">
<span class="plain-syntax">[ </span><span class="identifier-syntax">VM_SquareRoot</span><span class="plain-syntax"> </span><span class="identifier-syntax">num</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">0</span><span class="plain-syntax">;</span>
<span class="plain-syntax">];</span>

<span class="plain-syntax">[ </span><span class="identifier-syntax">VM_CubeRoot</span><span class="plain-syntax"> </span><span class="identifier-syntax">num</span><span class="plain-syntax">;</span>
<span class="plain-syntax">    </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">0</span><span class="plain-syntax">;</span>
<span class="plain-syntax">];</span>
</pre>
<p class="commentary firstcommentary"><a id="SP4" class="paragraph-anchor"></a><b>&#167;4. Real number support.</b>These need to exist, but are likely never to be used.
</p>

<pre class="displayed-code all-displayed-code code-font">
<span class="plain-syntax">[ </span><span class="identifier-syntax">REAL_NUMBER_TY_Say</span><span class="plain-syntax"> </span><span class="identifier-syntax">real</span><span class="plain-syntax">; </span><span class="reserved-syntax">print</span><span class="plain-syntax"> </span><span class="identifier-syntax">real</span><span class="plain-syntax">; ];</span>

<span class="plain-syntax">[ </span><span class="identifier-syntax">REAL_NUMBER_TY_Compare</span><span class="plain-syntax"> </span><span class="identifier-syntax">r1</span><span class="plain-syntax"> </span><span class="identifier-syntax">r2</span><span class="plain-syntax">; </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">UnsignedCompare</span><span class="plain-syntax">(</span><span class="identifier-syntax">r1</span><span class="plain-syntax">, </span><span class="identifier-syntax">r2</span><span class="plain-syntax">); ];</span>
</pre>
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